In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

Solution:

It is given that the question paper consists of 12 questions divided into two parts – Part I and Part II, containing 5 and 7 questions, respectively.

A student has to attempt 8 questions, selecting at least 3 from each part. 

This can be done in three ways as follows:

1. 3 questions from part I and 5 questions from part II
2. 4 questions from part I and 4 questions from part II
3. 5 questions from part I and 3 questions from part II

We know that the number of ways of selecting r different things from n different things is a combination and is calculated using the formula ⁿCᵣ = n! / [r!(n−r)!]. Using this,

• 3 questions from part I and 5 questions from part II can be selected in ⁵C₃ × ⁷C₅ ways.
• 4 questions from part I and 4 questions from part II can be selected in ⁵C₄ × ⁷C₄ ways.
• 5 questions from part I and 3 questions from part II can be selected in ⁵C₅× ⁷C₃ ways.

Thus, the required number of ways of selecting questions:

⁵C₃ × ⁷C₅ + ⁵C₄ × ⁷C₄+ ⁵C₅ × ⁷C₃

= [5!/(2!3!) × 7!/(5!2!) ] + [ 5!/(4!1!) × 7!/(4!3!)] + [ 5!/(1!5!) × 7!/(3!4!) ] (Using nCr formula)

= 210 + 175 + 35

= 420

NCERT Solutions Class 11 Maths Chapter 7 Exercise ME Question 7

In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions?

Summary:

If a student is required to attempt 8 questions in all, selecting at least 3 from each part., then the required number of ways of selecting questions is 420